Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model

We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of $\mathbb{Z}^d$ ($d\geq 2$) with zero Dirichlet condition. We assume that the conductances $w$ are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If $\gamma=\sup \{ q\geq 0\colon \mathbb{E} [w^{-q}] 1/4$ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.